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b/src/fundamentals-of-ai-and-kr/module1/main.tex @@ -11,5 +11,6 @@ \input{sections/_search.tex} \input{sections/_local_search.tex} \input{sections/_swarm_intelligence.tex} + \input{sections/_games.tex} \end{document} \ No newline at end of file diff --git a/src/fundamentals-of-ai-and-kr/module1/sections/_games.tex b/src/fundamentals-of-ai-and-kr/module1/sections/_games.tex new file mode 100644 index 0000000..e5d9020 --- /dev/null +++ b/src/fundamentals-of-ai-and-kr/module1/sections/_games.tex @@ -0,0 +1,88 @@ +\chapter{Games} + +In this course, we are interested in two-player games with +perfect knowledge (both players have the same information on the state of the game). + + +\section{Minimax algorithm} +\marginnote{Minimax algorithm} +The minimax (min-max) algorithm allows to determine the optimal strategy for a player by +building and propagating utility scores in a tree where each node represents a state of the game. +It considers the player as the entity that maximizes (\textsc{Max}) its utility and +the opponent as the entity that (optimally) minimizes (\textsc{Min}) the utility of the player. + + +\begin{figure}[h] + \centering + \includegraphics[width=0.5\textwidth]{img/_minmax.pdf} + \caption{Example of game tree with propagated scores} +\end{figure} + +In a game tree, each level represents the actions a that single player can do. +In minimax, the levels where the player plays are the \textsc{Max} levels, +while the levels of the opponent are the \textsc{Min} levels. + +Given a node $n$ of a game tree, +an iteration of the minimax algorithm can be described as follows: +\begin{descriptionlist} + \item[Expansion] + Expansion of the sub-tree having $n$ as root by considering the possible moves starting from the state of $n$. + + The expansion in height stops when a final state is reached or when some predetermined conditions are met + (note that different branches may be expanded with different heights). + + The expansion in width stops when all the possible moves have been considered or when some predetermined conditions are met. + + \item[Evaluation] + Each leaf is labeled with a score. + For terminal states, a possible score assignment is: + \[ \texttt{utility}(\texttt{state}) = \begin{cases} + +1 & \text{Win} \\ + -1 & \text{Loss} \\ + 0 & \text{Draw} + \end{cases} \] + For non-terminal states, heuristics are required. + + \item[Propagation] + Starting from the parents of the leaves, the scores are propagated upwards + by labeling the parents based on the children's score. + + Given an unlabeled node $m$, if $m$ is at a \textsc{Max} level, its label is the maximum of its children's score. + Otherwise (\textsc{Min} level), the label is the minimum of its children's score. +\end{descriptionlist} + + +\begin{center} + \def\arraystretch{1.2} + \begin{tabular}{c | m{10cm}} + \hline + \textbf{Completeness} & Yes, if the game tree is finite \\ + \hline + \textbf{Optimality} & Yes, assuming an optimal opponent (otherwise, it may need more moves) \\ + \hline + \textbf{Time complexity} + & $O(b^m)$, with breadth $m$ and branching factor $b$ \\ + \hline + \textbf{Space complexity} + & $O(b \cdot m)$, with breadth $m$ and branching factor $b$ \\ + \hline + \end{tabular} +\end{center} + +\begin{algorithm} +\caption{Minimax algorithm} +\begin{lstlisting}[mathescape=true] +def minimax(node, who_is_next): + if isLeaf(node): + eval = evaluate(node) + elif who_is_next == ME: + eval = $-\infty$ + for c in node.children: # Expansion if needed + eval = max(eval, minimax(c, OPPONENT)) + elif who_is_next == OPPONENT: + eval = $+\infty$ + for c in node.children: # Expansion if needed + eval = min(eval, minimax(c, ME)) + return eval +\end{lstlisting} +\end{algorithm} \ No newline at end of file